The meaning of Ashura is what is especially important to understand. Reuters actually had a story which I can no longer find relating Husayn's death in a dispute over political leadership to the current conflict among the Shi'ite ulama in Iraq. This is kind of like using the story of Jesus's trial before the Sanhedrin and relating it to the controversy among contemporary American Catholics over Senators who vote for abortion. You might be able to make something of it, but it doesn't really help you understand Christian or Jewish beliefs and doctrines.
The Umayyads in Islamic tradition are seen as perverting the early Islamic community into a kingdom. At that time, many looked to the family of the Prophet as people who could provide true moral leadership, but Husayn's brother Hassan had prior to his death agreed to accept Umayyad rule and concentrated on religious matters. Husayn began this way, too, but many sought in him someone who would stand up to the unjust Umayyad tyranny and restore Islamic values in the community. Husayn agreed, but most of those who promised him aid abandoned him, and his very small band of followers was massacred.
The significance of this comes on several levels. One is the fact that not only was Husayn a victim of tyranny who shows how the powers of the world are traditionally evil, but in the hagiography surrounding him, he becomes very similar to Jesus in the gospels in that he basically foresees everything that will happen to him. As told by Shi'ites, one of the important aspects of his life is that he went forward to resist Yazid knowing it would mean his death but believing it was important to cause people to remember the righteous way of living and set an example of self-sacrifice to bring it about. Also important are the Penitents - those who abandoned Husayn to his fate, who were the first to be affected by his sacrifice and quickly went to take up his struggle.
For Shi'ites, Ashura functions much like the Christian Good Friday or the Jewish Yom Kippur. It is basically a time of atonement and self-reflection, but because of Islam's concern with community involvement and reform can have as much public as private significance, especially since in many societies governments have crushed direct opposition, but a story like that of Husayn is an unavoidable part of Islam, and no dictator can crush it even if he feels uncomfortably like people condemning Yazid might be thinking of him. But whatever you see on TV, remember it comes ultimately from people's personal religious values as they consider this story. Do I have the courage of Husayn to stand for what is right knowing ill will befall me? Who are the Yazids of my world, both on the large and small level, and what am I doing to resist them? What things have I done for which I must repent, and how can I turn those sins into virtues during the rest of my life? These are the questions on the minds of Shi'ite pilgrims this day, questions which I think challenge all of us regardless of our religion. After all, as the saying goes: "For every day is Ashura, and every land Karbala."